Commutator methods for the spectral analysis of uniquely ergodic dynamical systems

TL;DR

This paper introduces a commutator-based method for analyzing the spectral properties of uniquely ergodic dynamical systems, demonstrating its effectiveness on various examples and improving existing results on horocycle flows.

Contribution

It develops a novel commutator approach for spectral analysis and applies it to time changes of horocycle flows, achieving broader conditions for absolute continuity.

Findings

Absolute continuity of spectra for certain dynamical systems

Weaker assumptions than previous literature for horocycle flows

Effective method applicable to various uniquely ergodic systems

Abstract

We present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows, we obtain absolute continuity under assumptions weaker than the ones to be found in the literature.